There are two digital filters of this type: FIR (Finite Impulse Response) type filters and IIR (Infinite Impulse Response) type filters. The digital filter of IIR-type feeds back an output to realize complicated frequency characteristics. That of FIR-type has simple structure and is widely used for low pass filtering, for example.
A digital filter is realized by DSP for applications requiring flexibility, or by special use circuit when high speed processing is necessary.
Neither filter type is able to realize both satisfactorily high speed processing and wide-usability.